Question

1 ∠ptv ≅ ∠rus given
2 overline{uv} ≅ overline{st} given
3 ∠p ≅ ∠r given
4 tv = uv + tu additive property of length
5 su = st + tu additive property of length
6 tv = st + tu substitution
7 su = tv
8 △ptv ≅ △rus
9 overline{pt} ≅ overline{ru}

Explanation:

Step1: Substitute \( SU = ST + TU \) and \( TV = ST + TU \)

Since \( SU = ST + TU \) (from step 5) and \( TV = ST + TU \) (from step 6), by the transitive property of equality, \( SU = TV \).

Step2: Prove \( \triangle PTV\cong\triangle RUS \)

We have \( \angle PTV\cong\angle RUS \) (given), \( \angle P\cong\angle R \) (given), and \( SU = TV \) (from step 7). By the AAS (Angle - Angle - Side) congruence criterion, \( \triangle PTV\cong\triangle RUS \).

Step3: Conclude \( \overline{PT}\cong\overline{RU} \)

Since \( \triangle PTV\cong\triangle RUS \), by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, \( \overline{PT}\cong\overline{RU} \).

Answer:

1. Transitive Property of Equality; 8. AAS (Angle - Angle - Side) Congruence Criterion; 9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)